Discrete Mathematics 2010-2018

Abstract:

It is well-known that the Pell equation $a^2-db^2=1$ in integers has a
non-trivial solution if and only if $d$ is positive and not a perfect
square. If one considers the polynomial analog, i.e., for fixed $D \in
\mathbb{C}[X]$, the equation $A^2-DB^2=1$, the matter is more
complicated. Indeed, for the existence of a solution the clear
necessary conditions that the degree of $D$ must be even and that $D$
cannot be a perfect square are not sufficient. While the case of
degree two is analogous to the integer case, there are non-square
polynomials of degree 4 such that the corresponding Pell equation is
not solvable. On the other hand, as in the integer case, once we have
a non-trivial solution, we have infinitely many and we call minimal
solution a solution $(A,B)$ with $A$ of minimal degree.

In joint work with Laura Capuano and Umberto Zannier we showed that
there exist equations $A^2-DB^2=1$, with $(A,B)$ minimal solution, for
any choice of degrees deg$D \geq 4$ even and deg$A \geq $deg$D/2$.

Abstract:

Abstract: It goes back to Lagrange that a real quadratic irrational
has always a periodic continued fraction. Starting from decades ago,
several authors proposed different definitions of a p-adic continued
fraction, and the definition depends on the chosen system of residues
mod p. It turns out that the theory of p-adic continued fractions has
many differences with respect to the real case; in particular, no
analogue of Lagranges theorem holds, and the problem of deciding
whether the continued fraction is periodic or not seemed to be not
known until now. In recent work with F. Veneziano and U. Zannier we
investigated the expansion of quadratic irrationals, for the p-adic
continued fractions introduced by Ruban, giving an effective criterion
to establish the possible periodicity of the expansion. This
criterion, somewhat surprisingly, depends on the ‘real’ value of the
p-adic continued fraction.

Abstract:

Abstract:


Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A \it local monoidal transform \rm of $R$ is a ring of the form $$R_1= R \left[ \frac{\mathfrak{p}}{x} \right]_{\mathfrak{m}_1}$$ where $ \mathfrak{p} $ is a prime ideal generated by regular parameters, $x \in \mathfrak{p}$ is a regular parameter and $ \mathfrak{m}_1 $ is a maximal ideal of $ R[\frac{\mathfrak{m}}{x}] $ lying over $ \mathfrak{m}. $ If $\mathfrak{p}= \mathfrak{m}$ the ring $R_1$ is called a \it local quadratic transform\rm.

Recently, several authors studied the rings of the form $ S= \cup_{n \geq 0}^{\infty} R_n $ obtained as infinite directed union of iterated local quadratic transforms of $R$, and call them \it quadratic Shannon extension\rm.
A directed union of local monoidal transforms of a regular local ring is said \it monoidal Shannon extension\rm.

Here we study features of monoidal Shannon extensions and more in general of directed unions of Noetherian UFDs.

(L. Guerrieri, Directed unions of local monoidal transforms and GCD domains (2018) arXiv:1808.07735 )

Abstract:

In the first part of the talk, I will speak about the normal structure of classical and classical-like groups. The description of the normal subgroups for various classes of concrete groups, and especially for classical groups over rings, has been one of the central themes of group theory in the last two centuries, right after Galois introduced the notion of normal subgroups. In the second part of the talk, I will speak about weighted Leavitt path algebras. Weighted Leavitt path algebras are algebras associated to weighted graphs. They generalise in a natural way the usual Leavitt path algebras and also Leavitt's algebras of module type $(n,k)$ where $n,k>0$.

Abstract:

This talk consists of two sections. In the first section,
we prove that if $D$ is an almost P$v$MD, then $D$ is of finite $t$-character
if and only if each nonzero $t$-locally finitely generated $w$-ideal of $D$ is of finite type.
As a corollary, we
get that if $D$ is an almost Pr\"ufer domain, then $D$ is of finite character
if and only if each locally finitely generated ideal of $D$ is finitely generated.

In the second section, we consider to
what extent conditions on the homogeneous elements or ideals of a graded integral domain $R$ carry
over to all elements or ideals of $R$.
For instance, we prove that $R$ is a gr-$t$-quasi-Pr\"ufer domain if and only if $R$ is a $t$-quasi-Pr\"ufer domain.
However, we show that homogeneously $tv$-domains and homogeneously $w$-divisorial domains are not equal to $tv$-domains and $w$-divisorial domains, respectively.


This talk is based on joint works with G. W. Chang and P. Sahandi.

Abstract:

Abstract: This talk contains two parts. In the first section of this talk, we define the set of integer-valued polynomials over the subsets of matrix rings. We do this on full matrix ring, upper triangular matrix ring and upper triangular matrix ring with constant diagonal. We present some examples to show that these sets may be not rings.
Then, we introduce some cases that the set of integer-valued polynomials over subsets of matrix ring is a ring. Furthermore, we consider some properties of these rings as Noetherian property and Krull dimension.
In the second section, we generalize the ring of integer-valued polynomials over upper triangular matrices and define the set of integer-valued polynomials over some cases of block matrices. Then, we show that this set is a ring. It solves the open problem of integer-valued polynomials on algebras in a special case of block matrix algebras.

Abstract:

\noindent Let $G$ be a group. A non-empty subset $S$ of $G$ is `product-free}' if $ab \not\in S$ for all $a, b \in S$. We call such a set `locally maximal}' in $G$ if it is not properly contained in any other product-free subset of $G$.
Locally maximal product-free sets were first studied in 1974 by Street and Whitehead, who analysed some properties of these sets, and introduced the concept of filled groups.
We say a locally maximal product-free subset $S$ of $G$ `fills}' $G$ if
$G^{\ast}\subseteq S \cup SS$ (where $G^{\ast}=G\setminus \{1\}$), and $G$ is called a `filled group}'
if every locally maximal product-free set in $G$ fills $G$.
In this talk, we shall consider questions like:
(a) for a given positive integer k, which finite groups contain a locally maximal product-free set of size k?; (b) how many finite groups are filled?.

Abstract:

\noindent
It is well known that Gauss Elimination produces a factorization into elementary matrices of any invertible matrix over a field. A classical problem, studied since the middle of the 1960's, is to characterize integral domains different from fields that satisfy the same property. As a partial answer, in 1993, Ruitenburg proved that in the class of B\'ezout domains, any invertible matrix can be written as a product of elementary matrices if and only if any singular matrix can be written as a product of idempotents.

\noindent
In this talk, after giving an overview of the classical results on these factorization properties, we will present some recent developments on the topic. In particular, we will consider products of elementary and idempotent matrices over special classes of non-Euclidean PID's and over integral domains that are not B\'ezout.

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\textsc{References}

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\begin{itemize}{\footnotesize

\item[\textnormal{[1]}] L.~Cossu, P.~Zanardo, U.~Zannier,
Products of elementary matrices and non-Euclidean principal ideal domains}, J. Algebra 501: 182–205, 2018.
\item[\textnormal{[2]}] L.~Cossu, P.~Zanardo,
Factorizations into idempotent factors of matrices over Pr\"ufer domains}, accepted for publication on Communications in Algebra, 2018.


}\end{itemize}

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\noindent{\footnotesize
\textsc{Department of Mathematics ``Tullio Levi Civita'', University of Padova}
{Via Trieste, 63}
{35121, Padova}
{Italy}
}

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\noindent{\footnotesize{E-mail address}:
lcossu@math.unipd.it}
}